Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid-viscous fluid interface. Tapering plates in either converging or diverging directions leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Simulations compare well with previous experimental results for various injection rates and geometric configurations. Further, we demonstrate how the fully nonlinear dynamics of the problem affect the number of fingers that emerge and how well this number agrees with predictions from linear stability analysis

Gas compression systematically delays the onset of viscous fingering

Using gas to drive liquid from a Hele-Shaw cell leads to classical viscous fingering. Strategies for suppressing fingering have received substantial attention. For steady injection of an incompressible gas, the intensity of fingering is controlled by the capillary number Ca. Here, we show that gas compression leads to an unsteady injection rate controlled primarily by a dimensionless compressibility number C. Increasing C systematically delays the onset of fingering at high Ca, highlighting compressibility as an overlooked but fundamental aspect of gas-driven fingering

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterised by a blow-up of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a novel numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent $\alpha = 1/3$ just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.Comment: 16 pages, 16 figure

Compression-driven viscous fingering in a radial Hele-Shaw cell

The displacement of a viscous liquid by a gas within a Hele-Shaw cell is a classical problem. The gas-liquid interface is hydrodynamically unstable, forming striking finger-like patterns that have attracted research interest for decades. Generally, both the gas and liquid phases are taken to be incompressible, with the capillary number being the key parameter that determines the severity of the instability. Here, we consider a radially outward displacement driven by the steady compression of a gas reservoir. The associated gas-injection rate is then unsteady due to the compressibility of the gas. We identify a second nondimensional parameter, the compressibility number, that plays a strong role in the development of the fingering pattern. We use an axisymmetric model to study the impact of compressibility number on the unsteady evolution of injection rate and gas pressure. We use linear stability analysis to show that increasing the compressibility number delays the onset of finger development relative to the corresponding incompressible case. Finally, we present and compare a series of experiments and fully nonlinear simulations over a broad range of capillary and compressibility numbers. These results show that increasing the compressibility number systematically decreases the severity of the fingering pattern at high capillary number. Our results provide an unprecedented comparison of experiments with simulations for viscous fingering, a comprehensive understanding of the role of compressibility in unstable gas-liquid displacement flows, and insight into a new mechanism for controlling the development of fingering patterns

Two-interface and thin filament approximation in Hele--Shaw channel flow

For a viscous fluid trapped in a Hele--Shaw channel, and pushed by a pressure difference, the fluid interface is unstable due to the Saffman--Taylor instability. We consider the evolution of a fluid region of finite extent, bounded between two interfaces, in the limit the interfaces are close, that is, when the fluid region is a thin liquid filament separating two gases of different pressure. In this limit, we derive a geometric flow rule that describes the normal velocity of the filament centreline, and evolution of the filament thickness, as functions of the thickness and centreline curvature. We show that transverse flow along the filament is necessary to regularise the instability. Numerical simulation of the thin filament flow rule is shown to closely match level-set computations of the complete two-interface model, and solutions ultimately evolve to form a bubble of increasing radius and decreasing thickness

Moving boundary problems for quasi-steady conduction limited melting

The problem of melting a crystal dendrite is modelled as a quasi-steady Stefan 5 problem. By employing the Baiocchi transform, asymptotic results are derived in the limit that 6 the crystal melts completely, extending previous results that hold for a special class of initial and 7 boundary conditions. These new results, together with predictions for whether the crystal pinches off 8 and breaks into two, are supported by numerical calculations using the level set method. The effects of 9 surface tension are subsequently considered, leading to a canonical problem for near-complete-melting 10 which is studied in linear stability terms and then solved numerically. Our study is motivated in 11 part by experiments undertaken as part of the Isothermal Dendritic Growth Experiment, in which 12 dendritic crystals of pivalic acid were melted in a microgravity environment: these crystals were 13 found to be prolate spheroidal in shape, with an aspect ratio initially increasing with time then 14 rather abruptly decreasing to unity. By including a kinetic undercooling-type boundary condition in 15 addition to surface tension, our model suggests the aspect ratio of a melting crystal can reproduce 16 the same non-monotonic behaviour as that which was observed experimentally. 1

A novel formulation of inhaled sodium cromoglicate (PA101) in idiopathic pulmonary fibrosis and chronic cough: a randomised, double-blind, proof-of-concept, phase 2 trial

Background Cough can be a debilitating symptom of idiopathic pulmonary fibrosis (IPF) and is difficult to treat. PA101 is a novel formulation of sodium cromoglicate delivered via a high-efficiency eFlow nebuliser that achieves significantly higher drug deposition in the lung compared with the existing formulations. We aimed to test the efficacy and safety of inhaled PA101 in patients with IPF and chronic cough and, to explore the antitussive mechanism of PA101, patients with chronic idiopathic cough (CIC) were also studied. Methods This pilot, proof-of-concept study consisted of a randomised, double-blind, placebo-controlled trial in patients with IPF and chronic cough and a parallel study of similar design in patients with CIC. Participants with IPF and chronic cough recruited from seven centres in the UK and the Netherlands were randomly assigned (1:1, using a computer-generated randomisation schedule) by site staff to receive PA101 (40 mg) or matching placebo three times a day via oral inhalation for 2 weeks, followed by a 2 week washout, and then crossed over to the other arm. Study participants, investigators, study staff, and the sponsor were masked to group assignment until all participants had completed the study. The primary efficacy endpoint was change from baseline in objective daytime cough frequency (from 24 h acoustic recording, Leicester Cough Monitor). The primary efficacy analysis included all participants who received at least one dose of study drug and had at least one post-baseline efficacy measurement. Safety analysis included all those who took at least one dose of study drug. In the second cohort, participants with CIC were randomly assigned in a study across four centres with similar design and endpoints. The study was registered with ClinicalTrials.gov (NCT02412020) and the EU Clinical Trials Register (EudraCT Number 2014-004025-40) and both cohorts are closed to new participants. Findings Between Feb 13, 2015, and Feb 2, 2016, 24 participants with IPF were randomly assigned to treatment groups. 28 participants with CIC were enrolled during the same period and 27 received study treatment. In patients with IPF, PA101 reduced daytime cough frequency by 31·1% at day 14 compared with placebo; daytime cough frequency decreased from a mean 55 (SD 55) coughs per h at baseline to 39 (29) coughs per h at day 14 following treatment with PA101, versus 51 (37) coughs per h at baseline to 52 (40) cough per h following placebo treatment (ratio of least-squares [LS] means 0·67, 95% CI 0·48–0·94, p=0·0241). By contrast, no treatment benefit for PA101 was observed in the CIC cohort; mean reduction of daytime cough frequency at day 14 for PA101 adjusted for placebo was 6·2% (ratio of LS means 1·27, 0·78–2·06, p=0·31). PA101 was well tolerated in both cohorts. The incidence of adverse events was similar between PA101 and placebo treatments, most adverse events were mild in severity, and no severe adverse events or serious adverse events were reported. Interpretation This study suggests that the mechanism of cough in IPF might be disease specific. Inhaled PA101 could be a treatment option for chronic cough in patients with IPF and warrants further investigation

Multiorgan MRI findings after hospitalisation with COVID-19 in the UK (C-MORE): a prospective, multicentre, observational cohort study

Introduction: The multiorgan impact of moderate to severe coronavirus infections in the post-acute phase is still poorly understood. We aimed to evaluate the excess burden of multiorgan abnormalities after hospitalisation with COVID-19, evaluate their determinants, and explore associations with patient-related outcome measures. Methods: In a prospective, UK-wide, multicentre MRI follow-up study (C-MORE), adults (aged ≥18 years) discharged from hospital following COVID-19 who were included in Tier 2 of the Post-hospitalisation COVID-19 study (PHOSP-COVID) and contemporary controls with no evidence of previous COVID-19 (SARS-CoV-2 nucleocapsid antibody negative) underwent multiorgan MRI (lungs, heart, brain, liver, and kidneys) with quantitative and qualitative assessment of images and clinical adjudication when relevant. Individuals with end-stage renal failure or contraindications to MRI were excluded. Participants also underwent detailed recording of symptoms, and physiological and biochemical tests. The primary outcome was the excess burden of multiorgan abnormalities (two or more organs) relative to controls, with further adjustments for potential confounders. The C-MORE study is ongoing and is registered with ClinicalTrials.gov, NCT04510025. Findings: Of 2710 participants in Tier 2 of PHOSP-COVID, 531 were recruited across 13 UK-wide C-MORE sites. After exclusions, 259 C-MORE patients (mean age 57 years [SD 12]; 158 [61%] male and 101 [39%] female) who were discharged from hospital with PCR-confirmed or clinically diagnosed COVID-19 between March 1, 2020, and Nov 1, 2021, and 52 non-COVID-19 controls from the community (mean age 49 years [SD 14]; 30 [58%] male and 22 [42%] female) were included in the analysis. Patients were assessed at a median of 5·0 months (IQR 4·2–6·3) after hospital discharge. Compared with non-COVID-19 controls, patients were older, living with more obesity, and had more comorbidities. Multiorgan abnormalities on MRI were more frequent in patients than in controls (157 [61%] of 259 vs 14 [27%] of 52; p<0·0001) and independently associated with COVID-19 status (odds ratio [OR] 2·9 [95% CI 1·5–5·8]; padjusted=0·0023) after adjusting for relevant confounders. Compared with controls, patients were more likely to have MRI evidence of lung abnormalities (p=0·0001; parenchymal abnormalities), brain abnormalities (p<0·0001; more white matter hyperintensities and regional brain volume reduction), and kidney abnormalities (p=0·014; lower medullary T1 and loss of corticomedullary differentiation), whereas cardiac and liver MRI abnormalities were similar between patients and controls. Patients with multiorgan abnormalities were older (difference in mean age 7 years [95% CI 4–10]; mean age of 59·8 years [SD 11·7] with multiorgan abnormalities vs mean age of 52·8 years [11·9] without multiorgan abnormalities; p<0·0001), more likely to have three or more comorbidities (OR 2·47 [1·32–4·82]; padjusted=0·0059), and more likely to have a more severe acute infection (acute CRP >5mg/L, OR 3·55 [1·23–11·88]; padjusted=0·025) than those without multiorgan abnormalities. Presence of lung MRI abnormalities was associated with a two-fold higher risk of chest tightness, and multiorgan MRI abnormalities were associated with severe and very severe persistent physical and mental health impairment (PHOSP-COVID symptom clusters) after hospitalisation. Interpretation: After hospitalisation for COVID-19, people are at risk of multiorgan abnormalities in the medium term. Our findings emphasise the need for proactive multidisciplinary care pathways, with the potential for imaging to guide surveillance frequency and therapeutic stratification

Physical, cognitive, and mental health impacts of COVID-19 after hospitalisation (PHOSP-COVID): a UK multicentre, prospective cohort study

Background The impact of COVID-19 on physical and mental health and employment after hospitalisation with acute disease is not well understood. The aim of this study was to determine the effects of COVID-19-related hospitalisation on health and employment, to identify factors associated with recovery, and to describe recovery phenotypes. Methods The Post-hospitalisation COVID-19 study (PHOSP-COVID) is a multicentre, long-term follow-up study of adults (aged ≥18 years) discharged from hospital in the UK with a clinical diagnosis of COVID-19, involving an assessment between 2 and 7 months after discharge, including detailed recording of symptoms, and physiological and biochemical testing. Multivariable logistic regression was done for the primary outcome of patient-perceived recovery, with age, sex, ethnicity, body-mass index, comorbidities, and severity of acute illness as covariates. A post-hoc cluster analysis of outcomes for breathlessness, fatigue, mental health, cognitive impairment, and physical performance was done using the clustering large applications k-medoids approach. The study is registered on the ISRCTN Registry (ISRCTN10980107). Findings We report findings for 1077 patients discharged from hospital between March 5 and Nov 30, 2020, who underwent assessment at a median of 5·9 months (IQR 4·9–6·5) after discharge. Participants had a mean age of 58 years (SD 13); 384 (36%) were female, 710 (69%) were of white ethnicity, 288 (27%) had received mechanical ventilation, and 540 (50%) had at least two comorbidities. At follow-up, only 239 (29%) of 830 participants felt fully recovered, 158 (20%) of 806 had a new disability (assessed by the Washington Group Short Set on Functioning), and 124 (19%) of 641 experienced a health-related change in occupation. Factors associated with not recovering were female sex, middle age (40–59 years), two or more comorbidities, and more severe acute illness. The magnitude of the persistent health burden was substantial but only weakly associated with the severity of acute illness. Four clusters were identified with different severities of mental and physical health impairment (n=767): very severe (131 patients, 17%), severe (159, 21%), moderate along with cognitive impairment (127, 17%), and mild (350, 46%). Of the outcomes used in the cluster analysis, all were closely related except for cognitive impairment. Three (3%) of 113 patients in the very severe cluster, nine (7%) of 129 in the severe cluster, 36 (36%) of 99 in the moderate cluster, and 114 (43%) of 267 in the mild cluster reported feeling fully recovered. Persistently elevated serum C-reactive protein was positively associated with cluster severity. Interpretation We identified factors related to not recovering after hospital admission with COVID-19 at 6 months after discharge (eg, female sex, middle age, two or more comorbidities, and more acute severe illness), and four different recovery phenotypes. The severity of physical and mental health impairments were closely related, whereas cognitive health impairments were independent. In clinical care, a proactive approach is needed across the acute severity spectrum, with interdisciplinary working, wide access to COVID-19 holistic clinical services, and the potential to stratify care. Funding UK Research and Innovation and National Institute for Health Research

A review of one-phase Hele-Shaw flows and a level-set method for nonstandard configurations

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.   doi:10.1017/S144618112100033